(iv) (x + y)s = xs + ys, for all x, y [member of] A and s [member
of] S, then we say that A is a distributively generated near-ring.

Definition 1.2. A near-ring (B, +, *) is Boolean-near-ring if there
exists a Boolean-ring (A, +, [and], 1) with identity such that is
defined in terms of +, [and] and 1, and for any b [member of] B, b * b =
b.

Definition 1.3. A near-ring (B, +, *) is said to be idempotent if
[x.sup.2] = x, for all x [member of] B. i.e. If (B, +, *) is an
idempotent ring, then for all a, b [member of] B, a + a = 0 and a * b =
b * a.

Definition 1.5. Let A = (..., a, b, c, ...) be a set of pairwise
compatible elements of an associate ring R. Let A be maximal in the
sense that each element of A is compatible with every other element of A
and no other such elements may be found in R. Then A is said to be a
maximal compatible set or a maximal set.

Definition 1.6. If a sub-direct sum R of domains has an identity,
and if R has the property that with each element a, it contains also the
associated idempotent [a.sup.0] of a, then R is called an associate
subdirect sum or an associate ring.

Definition 1.7. If the maximal set A contains an element u which
has the property that a < u, for all a [member of] A, then u is
called the uni-element of A.

Definition 1.9. A Boolean-near-ring (B, [disjunction], [and]) is
said to be Samarandache-Boolean-near-ring whose proper subset A is a
Boolean-ring with respect to same induced operation of B.

Theorem 1.1. A Boolean-near-ring (B, [disjunction], [and]) is
having the proper subset A, is a maximal set with uni-element in an
associate ring R, with identity under suitable definitions for (B, +,)
with corresponding lattices (A, [less than or equal to])(A, <) and

Proof. Given (B, [disjunction], [and]) is a Boolean-near-ring whose
proper subset (A, [disjunction], [and]) is a maximal set with
uni-element in an associate ring R, and if the maximal set A is also a
subset of B.

Now to prove that B is Smarandache-Boolean-near-ring. It is enough
to prove that the proper subset A of B is a Boolean-ring. Let a and b be
two constants of A, if a is compatible to b, we define a [and] b as
follows:

Similarly, the element a [and] b = a [intersection] b = [a.sup.0]b
= a[b.sup.0] = glb(a, b) has defined and shown to belongs to A as the
glb(a, b). Now let us show that (A, [disjunction], [and]) is a
Boolean-ring. Firstly, a [disjunction] a = 0, since [a.sub.i] =
[a.sub.i] in every i-component, whence [(a[disjunction]a).sub.i]
vanishes, by our definition of '[and]'. Secondly a [and] a = a
[intersection] a = [a.sup.0]a = a, and so a is idempotent under [and].
We have shown that A is closed under [and] is [disjunction], and
associativity is a direct verification, and each element is itself
inverse under [and].

To prove associativity under [and]:

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For distributivity of [and] over [and], let c be an arbitrary
element in A.

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence (A,
[disjunction], [and]) is a Boolean-ring.

It follows that the proper subset A, a maximal set of B forms a
Boolean ring. B is a Boolean-near-ring, whose proper subset is a
Boolean-ring, then by definition, B is a SmarandacheBoolean-near-ring.

Theorem 1.2. A Boolean-near-ring (B, [disjunction], [and]) is
having the proper subset (A, +, [and], 1) is an associate ring in which
the relation of compatibility is transitive for non-zero elements with
identity under suitable definitions for (B, +, *) with corresponding
lattices (A, [less than or equal to])(A, <) and

Proof. Assume that (B, +, *) be Boolean- near-ring having a proper
subset A is an associate ring in which the relation of compatibility is
transitive for non-zero elements.

Now to prove that B is a Smarandache-Boolean-near-ring, i.e., to
prove that if the proper subset of B is a Boolean-ring, then by
definition B is Smarandache-Boolean-near-ring. We have 0 is compatible
with all elements, whence all elements are compatible with A and
therefore, are idempotent.

Then assume that transitivity holds for compatibility of non-zero
elements. It follows that non-zero elements from two maximal sets cannot
be compatible (much less equal), and hence, except for the element 0,
the maximal sets are disjoint.

Let a be a arbitrary, non-zero element of R. If a is a zero-divisor
of R, then the idempotent element A - [a.sup.0] [not equal to] 0.
Further A - [a.sup.0] belongs to the maximal set generated by the
non-zero divisor a' = a + A - [a.sup.0], since it is
(A-[a.sup.0])a' = (A - [a.sup.0])(a + A-[a.sup.0]) = (A -
[a.sup.0]) = [(A - [a.sup.0]).sup.2] i.e. A - [a.sup.0] < a'.
Since also a < a' and a ~ A - [a.sup.0], therefore, a is
idempotent. i.e. All the zero-divisors of R are idempotent which is a
maximal set then by theorem 1 and by definition A is a Boolean-ring.
Then by definition, B is Smarandache-Boolean-near-ring.

Theorem 1.3. A Boolean-near-ring (B, [disjunction], [and]) is
having the proper subset A, the set A of idempotent elements of a ring
R, with suitable definitions for [disjunction] and [and],

Assume that the set A of idempotent elements of a ring R, which is
also a subset of B. Now to prove that B is a
Smarandache-Boolean-near-ring. It is sufficient to prove that the set A
of idempotent elements of a ring R with identity forms a maximal set in
R with uni-element. By the definition of compatible, then we have every
element of R is compatible with every other idempotent element. If a
[member of] R is not idempotent then, [a.sup.2] * 1 [not equal to] a *
[1.sup.2], since the definition of compatible. Hence no non-idempotent
can belong to this maximal set. Thus the set A is idempotent element of
R with identity forms a maximal set in R whose uni-element is the
identity of R, by theorem 1 and by definition. A, a maximal set of B
forms a Boolean ring

Then by definition, it concludes that B is
Smarandache-Boolean-near-ring.

Theorem 1.4. A Boolean-near-ring (B, [disjunction], [and]) is
having the proper subset, having a nonzero divisor of A, as an associate
ring, with suitable definitions for [disjunction] and [and],

Now to prove that B Smarandache-Boolean-near-ring. It is enough to
prove that every non-divisor of A determines uniquely a maximal set of A
with uni-element.

Let a be the uni-element of a maximal set A then we have b < a,
for b [member of] A.

Consider all the elements of A in whose sub-direct display one or
more component [a.sub.i] duplicate the corresponding component [u.sub.i]
of u, the other components of a being zeros, i.e., all the element a
such that a < u, becomes u is uni-element. Clearly, these elements
are compatible with each other and together with u form a maximal set in
A, for which u is the uni-element. Hence A is a maximal set with
uni-element and by theorem 1 and definition A, a maximal set of B forms
a Boolean ring.

Then by definition, B is Smarandache-Boolean-near-ring.

Theorem 1.5. A Boolean-near-ring (B, [disjunction], [and]) is
having the proper subset A, associate ring is of the form A = [u.sub.J],
where u is a non-zero of A and J is the set of idempotent elements of A,
with suitable definitions for [disjunction] and [and],

Proof. Assume that the proper subset A of a Boolean-near-ring B is
of the form A = [u.sub.J], where u is non-zero divisor of A and J is the
set of idempotent elements of A. Now to prove B is
Smarandache-Boolean-near-ring. It is enough to prove that A is a maximal
set with uni-element.

(i) It is sufficient to show that the set [u.sub.J] is a maximal
set having u as its uni-element.

(ii) If b belongs to the maximal set determined by u, then b has
the required form b = [u.sub.e], for some e [member of] J.

Proof of (i). It is seen that [u.sub.e] ~ [u.sub.f] i.e. [u.sub.e]
is compatible to [u.sub.f] with uni-element u, for all e, f [member of]
J, since idempotent belongs to the center of A. Also, [u.sub.e] < u,
since [u.sub.e] * u = [u.sup.2.sub.e] = [([u.sub.e]).sup.2].

Proof of (ii). We know that A is an associate ring, the associated
idempotent [a.sup.0] of a has the property: if a ~ b then [a.sup.0]b =
a[b.sup.0] = [b.sup.0]a = b[a.sup.0]; if a [member of] [A.sub.u], then
since a < u and [u.sup.0] = 1, we have A = [u.sup.0]a = a[u.sup.0] =
[a.sup.0]u, for all [a.sup.0] [member of] J.

Hence A is a maximal set with uni-element of of B by suitable
definition and by theorem 1 then we have A is a Boolean-ring. Then by
definition, B is Smarandache-Boolean-near-ring.